Then we prove asymptotic normality of \(\widehat{{\mathbf {\theta }}}_{mix\cdot em}\)  using Theorem 5.14 of [1]}. Since \(\mathbf {Y}_i\)  has finite \((4+\delta )\) -th moment, \(\sup _i\Vert \psi _i(\mathbf {Y}_i,\mathbf {\theta })\Vert ^{2+\frac{\delta }{2}}<\infty \) . Then, if conditions \(\liminf _n\lambda _-[n^{-1}\textnormal {Var}(s_n(\mathbf {\theta }))]>0\)  and \(\liminf _n\lambda _-[n^{-1}\mathbf {M}_n(\mathbf {\theta })]>0\)  holds, we have
\(\sqrt{n}(\widehat{{\mathbf {\theta }}}_{mix\cdot em} - \mathbf {\theta }_0)\overset{d}{\rightarrow }N(0, \widetilde{\mathbf {V}}),\) 
